metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊Q8⋊5C2, C4⋊C4.18D6, C24⋊C4⋊19C2, (C2×C8).173D6, (C2×Q8).34D6, Q8⋊C4⋊15S3, Q8⋊2Dic3⋊1C2, C6.D8.1C2, C4.29(C4○D12), C12.14(C4○D4), C2.15(Q8⋊3D6), C6.60(C8⋊C22), (C2×Dic3).26D4, C2.D24.10C2, C2.9(Q16⋊S3), C22.188(S3×D4), (C6×Q8).17C22, C4.55(D4⋊2S3), (C2×C12).234C23, (C2×C24).236C22, C12.23D4.2C2, C6.27(C4.4D4), (C2×D12).57C22, C6.54(C8.C22), C4⋊Dic3.84C22, (C4×Dic3).18C22, C3⋊3(C42.28C22), C2.17(C23.11D6), (C2×C6).247(C2×D4), (C2×C3⋊C8).29C22, (C3×Q8⋊C4)⋊20C2, (C3×C4⋊C4).35C22, (C2×C4).341(C22×S3), SmallGroup(192,353)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C8).D6
G = < a,b,c,d | a2=b8=1, c6=d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab3, dbd-1=ab-1, dcd-1=ac5 >
Subgroups: 328 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, C2×D12, C6×Q8, C42.28C22, C6.D8, C24⋊C4, C2.D24, Q8⋊2Dic3, C3×Q8⋊C4, C12⋊Q8, C12.23D4, (C2×C8).D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4.4D4, C8⋊C22, C8.C22, C4○D12, S3×D4, D4⋊2S3, C42.28C22, C23.11D6, Q8⋊3D6, Q16⋊S3, (C2×C8).D6
Character table of (C2×C8).D6
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 24 | 2 | 2 | 2 | 8 | 8 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2i | -2i | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | complex lifted from C4○D4 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -2i | 2i | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 2i | -2i | 0 | 0 | -1 | 1 | -√-3 | -√3 | √-3 | √3 | i | -i | -i | i | complex lifted from C4○D12 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -2i | 2i | 0 | 0 | -1 | 1 | √-3 | -√3 | -√-3 | √3 | -i | i | i | -i | complex lifted from C4○D12 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 2i | -2i | 0 | 0 | -1 | 1 | √-3 | √3 | -√-3 | -√3 | i | -i | -i | i | complex lifted from C4○D12 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -2i | 2i | 0 | 0 | -1 | 1 | -√-3 | √3 | √-3 | -√3 | -i | i | i | -i | complex lifted from C4○D12 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | √6 | -√6 | -√6 | orthogonal lifted from Q8⋊3D6 |
| ρ24 | 4 | 4 | 4 | 4 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | -√6 | √6 | √6 | orthogonal lifted from Q8⋊3D6 |
| ρ27 | 4 | 4 | -4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | √-6 | -√-6 | √-6 | complex lifted from Q16⋊S3 |
| ρ30 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | -√-6 | √-6 | -√-6 | complex lifted from Q16⋊S3 |
►On 96 points
Generators in S96
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 94 17 64 28 59 81 48)(2 71 82 89 29 43 18 54)(3 96 19 66 30 49 83 38)(4 61 84 91 31 45 20 56)(5 86 21 68 32 51 73 40)(6 63 74 93 33 47 22 58)(7 88 23 70 34 53 75 42)(8 65 76 95 35 37 24 60)(9 90 13 72 36 55 77 44)(10 67 78 85 25 39 14 50)(11 92 15 62 26 57 79 46)(12 69 80 87 27 41 16 52)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 38 7 44)(2 37 8 43)(3 48 9 42)(4 47 10 41)(5 46 11 40)(6 45 12 39)(13 53 19 59)(14 52 20 58)(15 51 21 57)(16 50 22 56)(17 49 23 55)(18 60 24 54)(25 69 31 63)(26 68 32 62)(27 67 33 61)(28 66 34 72)(29 65 35 71)(30 64 36 70)(73 92 79 86)(74 91 80 85)(75 90 81 96)(76 89 82 95)(77 88 83 94)(78 87 84 93)
G:=sub<Sym(96)| (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,94,17,64,28,59,81,48)(2,71,82,89,29,43,18,54)(3,96,19,66,30,49,83,38)(4,61,84,91,31,45,20,56)(5,86,21,68,32,51,73,40)(6,63,74,93,33,47,22,58)(7,88,23,70,34,53,75,42)(8,65,76,95,35,37,24,60)(9,90,13,72,36,55,77,44)(10,67,78,85,25,39,14,50)(11,92,15,62,26,57,79,46)(12,69,80,87,27,41,16,52), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,38,7,44)(2,37,8,43)(3,48,9,42)(4,47,10,41)(5,46,11,40)(6,45,12,39)(13,53,19,59)(14,52,20,58)(15,51,21,57)(16,50,22,56)(17,49,23,55)(18,60,24,54)(25,69,31,63)(26,68,32,62)(27,67,33,61)(28,66,34,72)(29,65,35,71)(30,64,36,70)(73,92,79,86)(74,91,80,85)(75,90,81,96)(76,89,82,95)(77,88,83,94)(78,87,84,93)>;
G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,94,17,64,28,59,81,48)(2,71,82,89,29,43,18,54)(3,96,19,66,30,49,83,38)(4,61,84,91,31,45,20,56)(5,86,21,68,32,51,73,40)(6,63,74,93,33,47,22,58)(7,88,23,70,34,53,75,42)(8,65,76,95,35,37,24,60)(9,90,13,72,36,55,77,44)(10,67,78,85,25,39,14,50)(11,92,15,62,26,57,79,46)(12,69,80,87,27,41,16,52), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,38,7,44)(2,37,8,43)(3,48,9,42)(4,47,10,41)(5,46,11,40)(6,45,12,39)(13,53,19,59)(14,52,20,58)(15,51,21,57)(16,50,22,56)(17,49,23,55)(18,60,24,54)(25,69,31,63)(26,68,32,62)(27,67,33,61)(28,66,34,72)(29,65,35,71)(30,64,36,70)(73,92,79,86)(74,91,80,85)(75,90,81,96)(76,89,82,95)(77,88,83,94)(78,87,84,93) );
G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(1,94,17,64,28,59,81,48),(2,71,82,89,29,43,18,54),(3,96,19,66,30,49,83,38),(4,61,84,91,31,45,20,56),(5,86,21,68,32,51,73,40),(6,63,74,93,33,47,22,58),(7,88,23,70,34,53,75,42),(8,65,76,95,35,37,24,60),(9,90,13,72,36,55,77,44),(10,67,78,85,25,39,14,50),(11,92,15,62,26,57,79,46),(12,69,80,87,27,41,16,52)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,38,7,44),(2,37,8,43),(3,48,9,42),(4,47,10,41),(5,46,11,40),(6,45,12,39),(13,53,19,59),(14,52,20,58),(15,51,21,57),(16,50,22,56),(17,49,23,55),(18,60,24,54),(25,69,31,63),(26,68,32,62),(27,67,33,61),(28,66,34,72),(29,65,35,71),(30,64,36,70),(73,92,79,86),(74,91,80,85),(75,90,81,96),(76,89,82,95),(77,88,83,94),(78,87,84,93)]])
Matrix representation of (C2×C8).D6 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 68 | 39 | 5 |
| 0 | 0 | 5 | 39 | 68 | 34 |
| 0 | 0 | 34 | 68 | 34 | 68 |
| 0 | 0 | 5 | 39 | 5 | 39 |
| 27 | 8 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 59 | 14 | 7 |
| 0 | 0 | 14 | 7 | 66 | 7 |
| 0 | 0 | 14 | 7 | 7 | 14 |
| 0 | 0 | 66 | 7 | 59 | 66 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 18 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 39 | 39 | 34 |
| 0 | 0 | 5 | 39 | 68 | 34 |
| 0 | 0 | 39 | 34 | 39 | 34 |
| 0 | 0 | 68 | 34 | 68 | 34 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,34,5,34,5,0,0,68,39,68,39,0,0,39,68,34,5,0,0,5,34,68,39],[27,0,0,0,0,0,8,46,0,0,0,0,0,0,66,14,14,66,0,0,59,7,7,7,0,0,14,66,7,59,0,0,7,7,14,66],[46,18,0,0,0,0,0,27,0,0,0,0,0,0,34,5,39,68,0,0,39,39,34,34,0,0,39,68,39,68,0,0,34,34,34,34] >;
(C2×C8).D6 in GAP, Magma, Sage, TeX
(C_2\times C_8).D_6
% in TeX
G:=Group("(C2xC8).D6"); // GroupNames label
G:=SmallGroup(192,353);
// by ID
G=gap.SmallGroup(192,353);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,1094,135,184,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^6=d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^3,d*b*d^-1=a*b^-1,d*c*d^-1=a*c^5>;
// generators/relations
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